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Question: How do you simplify \( \dfrac{{\tan x}}{{\sin x}} + \dfrac{1}{{\cos x}}? \)...

How do you simplify tan⁡xsin⁡x+1cos⁡x?\dfrac{{\tan x}}{{\sin x}} + \dfrac{1}{{\cos x}}?

Explanation

Solution

In this problem we have given a trigonometrically equation and we are asked to simplify the given trigonometrically equation. Here to solve the problem we need to use some trigonometric identities and need to substitute those identities in the given equation which helps us to simplify the given equation.

Formula used: tan⁡x=sin⁡xcos⁡x\tan x = \dfrac{{\sin x}}{{\cos x}}
1cos⁡x=sec⁡x\dfrac{1}{{\cos x}} = \sec x

Complete step-by-step solution:
It is given that tan⁡xsin⁡x+1cos⁡x\dfrac{{\tan x}}{{\sin x}} + \dfrac{1}{{\cos x}}
First we simplify each term.
Rewrite tan⁡x\tan x in terms of sines and cosines.
We can use tan⁡x=sin⁡xcos⁡x\tan x = \dfrac{{\sin x}}{{\cos x}} in the given equation, we get
⇒sin⁡xcos⁡xsin⁡x+1cos⁡x−−−−−(1)\Rightarrow \dfrac{{\dfrac{{\sin x}}{{\cos x}}}}{{\sin x}} + \dfrac{1}{{\cos x}} - - - - - (1)
Now we can rewrite sin⁡xcos⁡xsin⁡x\dfrac{{\dfrac{{\sin x}}{{\cos x}}}}{{\sin x}} as a product form.
So, equation (1) becomes,
⇒sin⁡xcos⁡x×1sin⁡x+1cos⁡x−−−−−(2)\Rightarrow \dfrac{{\sin x}}{{\cos x}} \times \dfrac{1}{{\sin x}} + \dfrac{1}{{\cos x}} - - - - - (2)
Now we can cancel the common factors of sin⁡x\sin x in the equation (2), then the equation becomes
⇒1cos⁡x×1+1cos⁡x\Rightarrow \dfrac{1}{{\cos x}} \times 1 + \dfrac{1}{{\cos x}} , next we can rewrite the expression.
⇒1cos⁡x+1cos⁡x\Rightarrow \dfrac{1}{{\cos x}} + \dfrac{1}{{\cos x}} , here the denominators are the same. So we can add the numerators and it becomes,
2cos⁡x\dfrac{2}{{\cos x}} , here we can rewrite 2cos⁡x\dfrac{2}{{\cos x}} as 2×1cos⁡x\dfrac{{2 \times 1}}{{\cos x}} and this can be written as 2×1cos⁡x2 \times \dfrac{1}{{\cos x}} .
Next we have to convert 1cos⁡x\dfrac{1}{{\cos x}} as sec⁡x\sec x
Now, 2×1cos⁡x2 \times \dfrac{1}{{\cos x}} becomes 2sec⁡x2\sec x .

Therefore, we simplified addition of two terms into a single term that is we simplified the given equation tan⁡xsin⁡x+1cos⁡x\dfrac{{\tan x}}{{\sin x}} + \dfrac{1}{{\cos x}} as 2sec⁡x2\sec x .

Note: Simplification is the process of replacing a mathematical expression by equivalence one that is simpler or usually shorter, for example Simplification of a fraction to an irreducible fraction, simplification of expressions, in computer algebra and simplification by conjunction elimination in inference in logic yields a simpler, but generally non- equivalence.
We have simplified this trigonometric problem by just simply substituting some trigonometric identities. In this problem, we have substituted the trigonometric identities in two places which helps us to simplify the problem and students need to be careful about the identities.